Plebanski Formulation of General Relativity: A Practical Introduction

We give a pedagogical introduction into an old, but unfortunately not very well-known formulation of GR in terms of self-dual two-forms due to Plebanski. Our presentation is rather explicit in that we show how the familiar textbook solutions: Schwarzschild, Volkoff-Oppenheimer, as well as those describing the Newtonian limit, graviton and homogeneous isotropic Universe can be obtained within this formalism. Our description shows how Plebanski formulation gives quite an economical alternative to the usual metric and frame-based schemes for deriving Einstein equations. [email protected] 1 Plebanski formulation of general relativity The aim of this short paper is to give a description of Plebanski formulation [1] of general relativity (GR) in a version that we found most suited for practical computations. Our presentation is very explicit, in that the standard textbook solutions of GR are obtained. As we shall see, given an antsatz for the metric, Plebanski formulation produces Einstein equations even more quickly than the already efficient tetrad method. In our opinion, the efficiency and beauty of this formulation may warrant its inclusion in general relativity textbooks. Our convention for the signature is (−,+,+,+). We start with a description of the Plebanski version of Einstein equations in the usual tensor notations. 1.1 Einstein condition and the Hodge operator Given a spacetime metric gμν the condition that this metric is Einstein reads: Rμν ∼ gμν , where Rμν := R ρ μ νρ is the Ricci tensor, and the fact that the proportionality coefficient in this condition must be a constant is implied by the (differential) Bianchi identity ∇Gμν = 0, where ∇μ is the metric-compatible derivative operator and Gμν = Rμν − (1/2)gμνR is the Einstein tensor. As usual, the quantity R is the Ricci scalar R := R μ, and all indices are raised and lowered with the metric. The Plebanski formulation of GR is based on the following simple and well-known reformulation of the Einstein condition in terms of the Hodge operator. Thus, let us introduce the operation of Hodge dual that acts on bivectors (anti-symmetric rank two tensors) Aμν : Aμν → ∗Aμν = 1 2 ǫ ρσ μν Aρσ, (1) where the quantity ǫμνρσ is the volume 4-form for the metric gμν . The following elementary properties of the Hodge operator are easily verified: its square is minus one and it is invariant under conformal transformations of the metric g → Ωg. Given the Riemann curvature tensor Rμνρσ one can apply the Hodge operator to either the first or the second pair of indices: ∗Rμνρσ := 1 2 ǫ μ ν μν Rμ′ν′ρσ, R ∗ μνρσ := 1 2 Rμνρ′σ′ǫ ρσ ρσ. (2) It is a straightforward computation to check that the Einstein condition (together with the first Bianchi identity Rμ[νρσ] = 0) is equivalent to the condition that the left and right Hodge duals of the Riemann tensor coincide: Rμν ∼ gμν ⇐⇒ ∗Rμνρσ = R μνρσ. (3)